zbMATH — the first resource for mathematics

Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure. (English) Zbl 0806.28004
The author considers \(n\)-rectifiable metric spaces \((X,\rho)\). He shows analogously to the Euclidean setting: For almost all \(x\in X\) with respect to the \(n\)-dimensional Hausdorff measure \({\mathcal H}^ n_ \rho\) (understood as in Federer’s book) there exist a norm \(\|\cdot\|_ x\) on \(\mathbb{R}^ n\), a map \(\phi_ x: X\to \mathbb{R}^ n\), and a closed set \(A_ x\subset X\) such that \(\phi_ x(x)= 0\), \[ \lim_{r\to 0} (\alpha(n) r^ n)^{-1} {\mathcal H}^ n_ \rho(B_ \rho(x,r)\cap A_ x)= 1, \] and \[ \limsup_{r\to 0} \left\{\left|1- {\|\phi_ x(y)- \phi_ x(z)\|_ x\over \rho(y,z)}\right|;\;y,z\in A_ x\cap B_ \rho(x,r),\;y\neq z\right\}= 0. \] He gives also an extension of ideas due to Kolmogoroff concerning maximal and minimal strongly metrically invariant measures. He has found an integral representation of these measures on the space \(C([0,1])\). As a consequence, the equivalence class of Banach-Mazur compactum containing \(\|\cdot\|_ x\) consists of all norms on \(\mathbb{R}^ n\) induced by an inner product provided that the maximal and minimal Kolmogoroff measures coincide.

28A78 Hausdorff and packing measures
26B10 Implicit function theorems, Jacobians, transformations with several variables
Full Text: DOI
[1] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147 – 190. · Zbl 0342.46034
[2] S. Banach and S. Mazur, Zur Theorie der linearen Dimension, Studia Math. 4 (1933), 100-112. · JFM 59.1075.01
[3] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. · Zbl 0633.53002
[4] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[5] William B. Johnson, Joram Lindenstrauss, and Gideon Schechtman, Extensions of Lipschitz maps into Banach spaces, Israel J. Math. 54 (1986), no. 2, 129 – 138. · Zbl 0626.46007 · doi:10.1007/BF02764938 · doi.org
[6] A. Kolmogoroff, Beiträge zur Maßtheorie, Math. Ann. 107 (1932), 351-366. · Zbl 0006.05001
[7] J. Lukeš, J. Malý, and L. Zajíček, Fine topological methods in real analysis and potential theory, Lecture Notes in Math., vol. 1189, Springer, New York, 1986. · Zbl 0607.31001
[8] Pertti Mattila, Hausdorff \? regular and rectifiable sets in \?-space, Trans. Amer. Math. Soc. 205 (1975), 263 – 274. · Zbl 0274.28004
[9] Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. · Zbl 0698.46008
[10] D. Preiss and J. Tišer, On Besicovitch \( \frac{1}{2}\)-problem, J. London Math. Soc. (2) 45 (1992), 279-287. · Zbl 0762.28003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.